A CFD Model for the Direct Coupling of the Combustion Process and Glass Melting Flow Simulation in Glass Furnaces

Abstract

The objectives of reducing and increasing pollutant emissions during the glass production process also apply to the glass industry, meaning that the accurate modeling of a glass furnace is of critical strategic value. In the available literature, several CFD studies have proposed various models with different levels of complexity. Two basic aspects are shared by the existing models, limiting their accuracy and their impact on furnace design: the combustion space is usually solved with reliance on simplified models (e.g., Flamelet and global kinetic mechanisms); and the glass tank is solved separately, using an iterative approach to couple two (or more) simulated domains. This work presents the development of an innovative CFD model to overcome these limitations and to perform accurate simulations of industrial glass furnaces. The reactive flow is solved using a reduced chemical kinetic mechanism and the EDC (eddy dissipation concept) turbulence–chemistry interaction model to properly reproduce the complex combustion development. The glass bath is solved as a laminar flow with the appropriate temperature-dependent glass properties. The two domains are simulated simultaneously and thermally coupled through an interface. This procedure allows for the more accurate calculation of the heat flow and the temperature distributions on the glass bath, accounting for their subsequent influence on the glass convective motions. The simulation of an existing glass furnace, along with selected comparisons with experimental data, are presented to demonstrate the validity of the proposed model. The developed model provides a contribution that allows us to advance the wider understanding of glass furnace dynamics.

1. Introduction

Glass manufacturing is an important industrial sector, having an impact on a variety of fields, such as the automotive, construction, and consumer goods industries. The efficiency and performance of a glass production plant are critical factors in determining its overall sustainability, distinguishing it from other industries like plastics. Remarkably, a glass furnace requires a considerable energy input as the raw material must be heated beyond 1250 °C for the melting process. Recent estimates within the European Union state that the glass industry consumes an average of 7.8 GJ per year, with emissions of 0.58 tons of CO₂ for every ton of saleable product produced [1]. In today’s global context, characterized by increasingly stringent regulations on energy consumption and greenhouse gas emissions, optimizing glass production systems has become a crucial strategic priority. A standard glass furnace comprises a combustion space and a glass tank, where turbulent diffusive flames, produced in the former, supply energy to the latter, mainly through radiation.

Raw materials are introduced into the glass tank, where they undergo chemical reactions, melting, and homogenization over several hours before leaving the furnace. Several methods have been developed to reduce the environmental footprint of glass production. These include the use of regenerative or recuperative chambers [2], the use of systems for pre-heating the raw materials [3], exhaust gas recirculation techniques [4], and air staging methods to reduce specific emissions such as the emission of NOx [5,6]. Moreover, advancements in glass bath technology involve the integration of boilers [7] and electrodes [8], enhancing the homogenization process and improving quality with reduced thermal energy requirements. Responding to evolving demands, the industry has increasingly embraced alternative fuels, often with higher hydrogen content [9,10]. This multifaceted approach underscores the sector’s commitment to environmental responsibility and energy efficiency.

The complexity of turbulent diffusive combustion, i.e., the related interactions with the flow field and the complex glass melting process, necessitates an adequate understanding of the underlying physical phenomena, demanding the gradual improvement of the CFD model. A rich body of literature has emerged in this domain, dating back to the latter half of the previous century. McConnels and Goodson [11] and Mase and Oda [12] introduced simplified 3D furnace models. Carvalho [13] developed a comprehensive furnace model with sub-models dedicated to the combustion space, glass tank, and batch blanket. This seminal work provided the basis for many subsequent advances where, in particular, glass bath modeling is developed [14,15,16,17,18,19,20,21,22,23]. Reviews are provided by [24,25,26,27,28,29].

Recently, three-dimensional simulations of glass furnaces have been carried out using commercial software. For instance, Abbassi and Khoshmanesh [30] modeled and experimentally validated a side-port glass furnace, breaking it down into three key components, namely, the combustion chamber, the batch blanket, and the glass tank, using Gambit/Fluent. Li et al. [31] applied glass furnace model software from Glass Service (Czech Republic) to create a 3D simulation of a glass bath combustion chamber. Meanwhile, Raic et al. [32] and Daurer et al. [33] used Ansys Fluent to explore oxy-fuel glass furnaces. These studies reflect continued efforts to improve both our understanding and the optimization of the complex processes occurring in glass furnaces.

Two basic characteristics can be found in almost all of the studies: the combustion model relies on simplified methods (e.g., Flamelet, global, or quasi-global kinetic mechanisms) to reduce the computational cost; and the combustion and the glass domains are solved separately, using an iterative procedure for their coupling. These features significantly influence combustion development, the distribution of heat flow, and the temperature on the glass surface and its subsequent impact on the glass bath. As an example, combustion models based on the hypothesis of infinitely fast kinetics are intrinsically unsuitable to reproduce the complex development of flames strongly interacting with combustion products. On the coupling side, the iterative procedure does not allow for the passing of both the heat flow and the temperature distributions, from one domain to the other, at the same time. The heat flow distribution computed from the combustion domain is then used as a boundary condition for the glass domain, while the temperature distribution resulting from the glass domain simulation provides the boundary condition for the combustion chamber.

Temperatures and heat flows are iteratively compared until convergence is achieved; however, this coupling method has inherent issues. Specifically, as pointed out by Abbassi and Khoshmanesh [30], due to the “missing information” (i.e., the heat flow or the temperature, depending on the domain considered), it is not possible to obtain the correspondence between the regions of maximum temperature and of maximum heat flow, which is typical of real furnaces. Additionally, the coupling process may require numerous iterations between the combustion space and the glass tank–batch blanket models. These challenges highlight the complexity involved in achieving a realistic and accurate simulation of the interplay between the combustion chamber and the glass bath in a furnace.

The research efforts of our group have played a significant role in advancing fluid dynamic models for various components of glass production plants. Specifically, the activity has covered the modeling of regenerative chambers [34,35], raw material preheating systems for glass [36], exhaust gas recirculation systems [37,38], air staging techniques [39], and the glass bath itself [40]. In terms of combustion, the focus has included investigating the fluid dynamics related to burners [41] and validating combustion models using established test cases from the literature [42,43]. Drawing on the accumulated expertise, a CFD model has been developed for the accurate simulation of an industrial glass furnace. The combustion space is simulated using a reduced kinetic mechanism together with the EDC turbulence–chemistry interaction model. The full coupling of the combustion space and the glass tank is adopted, ensuring congruence on the glass surface between the heat flow and the temperature distributions. The model has been used to simulate an existing glass furnace, whose main characteristics are provided in the initial section of this work, which is followed by a description of the numerical model. The subsequent presentation unveils the main outcomes derived from the model, along with comparisons of the numerical results against the experimental measurements. The model can achieve remarkable accuracy in reproducing the actual combustion process, as well as the glass convective motions. The primary objective of this model is to develop an accurate tool for the design of glass furnaces, enabling complex parametric studies based on varying fuel types and geometric parameters. This capability enhances our ability to refine and optimize furnace designs for improved efficiency and reduced environmental impact.

2. Furnace

This study focuses on the simulation of a glass production plant specifically designed for the manufacturing of pharmaceutical-grade borosilicate glass. The furnace is a gas-fired, recuperative, end-port system with compact dimensions. The combustion chamber features an arc-shaped crown and includes two air vents, each with two burners, generating two primary flames simultaneously. Natural gas is introduced through fuel jets at a temperature of 300 K, while pre-heated air from the recovery chamber is supplied via the air vents, maintaining a 5% deficit air ratio relative to the gas flow rate. The exhaust gas is vented through a port located above the two air vents.

The glass tank, designed for low-volume molten glass production, receives raw materials through a single doghouse on the right side, which includes a portion of recycled material at ambient temperature. Due to the specific design of the plant, the batch blanket is restricted to a small area near the doghouse. To separate the melting zone from the refining zone, a weir wall is positioned along the tank floor at approximately two-thirds of the tank’s length. The glass tank is equipped with electrodes and boilers, though they are currently inactive.

Figure 1 presents a schematic view of a recuperative glass furnace.

3. Mathematical Model

3.1. Combustion Space

The combustion development is highly complex due to the interplay of chemical reactions, turbulence, and radiative heat transfer. The models used to account for these phenomena are briefly outlined below.

The physical–mathematical model relies on Reynolds-averaged equations, where each dependent variable ($\Phi$) is expressed as the sum of its mean and fluctuating components, and each balance equation is averaged according to the Reynolds rules.

$$\Phi =\overline{\Phi }+{\Phi }^{\prime }; \overline{{\Phi }^{\prime }}=0$$

(1)

3.1.1. Fluid Dynamics Governing Equations

The averaged balance equations for the mass, momentum, and total energy, expressed in differential form, are as follows:

$${\displaystyle \frac{\partial \overline{\rho }}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \overline{\rho }\overline{{\mathrm{u}}_{\mathrm{i}}}}{\partial {\mathrm{x}}_{\mathrm{i}}}}=0$$

(2)

$${\displaystyle \frac{\partial \overline{\rho }\overline{{\mathrm{u}}_{\mathrm{i}}}}{\partial \mathrm{t}}}+{\displaystyle \frac{\partial \overline{\rho }\overline{{\mathrm{u}}_{\mathrm{i}}}\overline{{\mathrm{u}}_{\mathrm{j}}}}{\partial {\mathrm{x}}_{\mathrm{j}}}}=-{\displaystyle \frac{\partial \overline{\mathrm{p}}}{\partial {\mathrm{x}}_{\mathrm{i}}}}+{\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}\left(\overline{{\mathrm{t}}_{\mathrm{ij}}-\rho {\mathrm{u}}_{\mathrm{i}}^{\prime }{\mathrm{u}}_{\mathrm{j}}^{\prime }}\right)$$

(3)

$$\overline{\rho }{\displaystyle \frac{\partial \left(\overline{\mathrm{e}}+\frac{1}{2}\overline{{\mathrm{u}}_{\mathrm{i}}}\overline{{\mathrm{u}}_{\mathrm{i}}}\right)}{\partial \mathrm{t}}}+\overline{\rho }{\displaystyle \frac{\partial \left(\overline{\mathrm{e}}+\frac{1}{2}\overline{{\mathrm{u}}_{\mathrm{i}}}\overline{{\mathrm{u}}_{\mathrm{i}}}\right)\overline{{\mathrm{u}}_{\mathrm{j}}}}{\partial {\mathrm{x}}_{\mathrm{j}}}}={\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}\left({\mathrm{k}}_{\mathrm{eff}}{\displaystyle \frac{\partial \mathrm{T}}{\partial {\mathrm{x}}_{\mathrm{j}}}}-\sum _{n=1}^N\overline{{\mathrm{h}}_{\mathrm{n}}}{\mathbf{J}}_{\mathbf{n}}\right)+{\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{j}}}}\overline{{\mathrm{u}}_{\mathrm{i}}}\left(-\mathrm{p}{\delta }_{\mathrm{ij}}+\overline{{\mathrm{t}}_{\mathrm{ij}}}-\overline{\rho {\mathrm{u}}_{\mathrm{i}}^{\prime }{\mathrm{u}}_{\mathrm{j}}^{\prime }}\right)+{\mathrm{S}}_{\mathrm{h}}$$

(4)

where the independent variables are the coordinates x_i and the time t. The subscript i = 1, 2, and 3 denotes the components of the Cartesian coordinates, corresponding to the x-axis (furnace width), y-axis (furnace length), and z-axis (furnace height), respectively, as illustrated in Figure 2. The summation convention for repeated indices is adopted.

The dependent variables u, p, T, e, and $\rho$ refer to the velocity, pressure, temperature, internal energy, and density; ${\mathrm{k}}_{\mathrm{eff}}$ is the sum of the molecular and turbulent thermal conductivities ($\mathrm{k}$ and ${\mathrm{k}}_{\mathrm{t}}$); ${\mathrm{S}}_{\mathrm{h}}$ is the source term, including the chemical heat of the reaction and the contribution due to the radiation mechanism; and ${\mathrm{t}}_{\mathrm{ij}}$ is the viscous stress tensor, modeled by the following:

$${\mathrm{t}}_{\mathrm{ij}}=2\mu {\mathrm{S}}_{\mathrm{ij}}$$

(5)

$${\mathrm{S}}_{\mathrm{ij}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial \overline{{\mathrm{u}}_{\mathrm{i}}}}{\partial {\mathrm{x}}_{\mathrm{j}}}}+{\displaystyle \frac{\partial \overline{{\mathrm{u}}_{\mathrm{j}}}}{\partial {\mathrm{x}}_{\mathrm{i}}}}\right)-{\displaystyle \frac{1}{3}}\mu {\mathrm{S}}_{\mathrm{kk}}{\delta }_{\mathrm{ij}}$$

(6)

where ${\mathrm{h}}_{\mathrm{n}}$ and ${\mathbf{J}}_{\mathbf{n}}$ are the sensible enthalpy and the diffusion flux of species n, respectively, where the following applies:

$${\mathbf{J}}_{\mathbf{n}}=-\left(\overline{\rho }{\mathrm{D}}_{\mathrm{m},\mathrm{n}}+{\mathrm{D}}_{\mathrm{t}}\right){\displaystyle \frac{\partial \overline{{\mathrm{Y}}_{\mathrm{n}}}}{\partial {\mathrm{x}}_{\mathrm{j}}}}$$

(7)

where ${\mathrm{Y}}_{\mathrm{n}}$ and ${\mathrm{D}}_{\mathrm{m},\mathrm{n}}$ are, respectively, the mass fraction and the molecular diffusion coefficient of species n; and ${\mathrm{D}}_{\mathrm{t}}$ is the corresponding turbulent diffusion coefficient.

3.1.2. Turbulence Model

The turbulence closure chosen for modeling the average effect of turbulent fluctuations is the standard k-ε model [44,45] with standard wall functions. It is known for delivering robust results and exhibiting high numerical stability, particularly in the presence of turbulent jets and reactive flows within industrial components [46].

The Reynolds stress tensor $\overline{\rho {\mathrm{u}}_{\mathrm{i}}^{\prime }{\mathrm{u}}_{\mathrm{j}}^{\prime }}$ is modeled with Boussinesq-type approximation.

$$\overline{\rho {\mathrm{u}}_{\mathrm{i}}^{\prime }{\mathrm{u}}_{\mathrm{j}}^{\prime }}={\displaystyle \frac{2}{3}}\stackrel{\mathrm{-}}{\rho }\mathrm{K}{\delta }_{\mathrm{ij}}-{\mu }_{\mathrm{t}}{2\mathrm{S}}_{\mathrm{ij}}$$

(8)

The turbulent viscosity (${\mu }_{\mathrm{t}}$) is computed by combining the turbulent kinetic energy (K) and its dissipation rate (ε).

$${{\mu }_{\mathrm{t}}=\mathrm{C}}_{\mu }\overline{\rho }{\displaystyle \frac{{\mathrm{K}}^2}{\epsilon }}$$

(9)

The two turbulent scales (i.e., K and ε) are obtained by solving the corresponding transport equations:

$${\displaystyle \frac{\partial \overline{\rho }\mathrm{K}}{\partial \mathrm{t}}}+{\overline{\mathrm{u}}}_{\mathrm{i}}{\displaystyle \frac{\partial \overline{\rho }\mathrm{K}}{\partial {\mathrm{x}}_{\mathrm{i}}}}=-\overline{\rho {\mathrm{u}}_{\mathrm{i}}^{\prime }{\mathrm{u}}_{\mathrm{j}}^{\prime }}{\mathrm{S}}_{\mathrm{ij}}+\left(\mu +{\displaystyle \frac{{\mu }_{\mathrm{T}}}{{\sigma }_{\mathrm{k}}}}\right){\displaystyle \frac{\partial \mathrm{K}}{\partial {\mathrm{x}}_{\mathrm{i}}}}-\overline{\rho }\epsilon$$

(10)

$${\displaystyle \frac{\partial \overline{\rho }\epsilon }{\partial \mathrm{t}}}+{\overline{\mathrm{u}}}_{\mathrm{i}}{\displaystyle \frac{\partial \overline{\rho }\epsilon }{\partial {\mathrm{x}}_{\mathrm{i}}}}=-{\mathrm{c}}_{\epsilon 1}{\displaystyle \frac{\epsilon }{\mathrm{K}}}\overline{\rho {\mathrm{u}}_{\mathrm{i}}^{\prime }{\mathrm{u}}_{\mathrm{j}}^{\prime }}{\mathrm{S}}_{\mathrm{ij}}+{\displaystyle \frac{\partial }{\partial {\mathrm{x}}_{\mathrm{i}}}}(\mu +{\displaystyle \frac{{\mu }_{\mathrm{T}}}{{\sigma }_{\epsilon }}}{\displaystyle \frac{\partial \epsilon }{\partial {\mathrm{x}}_{\mathrm{i}}}})-{\mathrm{c}}_{\epsilon 2}\overline{\rho }{\displaystyle \frac{{\epsilon }^2}{\mathrm{K}}}$$

(11)

where the model coefficients are set in accordance with Jones and Launder [47].

The turbulent thermal conductivity and the species turbulent diffusion are modeled via the following:

$${\mathrm{k}}_{\mathrm{t}}={\displaystyle \frac{{{\mu }_{\mathrm{t}}\mathrm{C}}_{\mathrm{P}}}{{\mathrm{Pr}}_{\mathrm{t}}}}$$

(12)

$${\mathrm{D}}_{\mathrm{t}}={\displaystyle \frac{{\mu }_{\mathrm{t}}}{{\mathrm{Sc}}_{\mathrm{t}}}}$$

(13)

where the turbulent Prandtl number (${\mathrm{Pr}}_{\mathrm{t}}$) is 0.85 and the Schmidt number (${\mathrm{Sc}}_{\mathrm{t}}$) is 0.7.

3.1.3. Radiation Model

The radiative heat transfer is computed using the P1 model, which is a simplification of the radiation transport equation, where the radiation intensity is assumed to be isotropic [48]. This model is coupled with the weighted-sum-of-grey-gases model (WSGGM) to account for spectral gas absorption effects.

3.1.4. Combustion Model

The species composition is obtained by solving the related balance equations, ex-pressed in the following form:

$${\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left(\overline{\rho }\overline{{\mathrm{Y}}_{\mathrm{n}}}\right)+{\displaystyle \frac{\partial \overline{\rho }\overline{{\mathrm{Y}}_{\mathrm{n}}}\overline{{\mathrm{u}}_{\mathrm{j}}}}{\partial {\mathrm{x}}_{\mathrm{j}}}}=-{\displaystyle \frac{\partial {\mathbf{J}}_{\mathbf{n}}}{\partial {\mathrm{x}}_{\mathrm{j}}}}+{\mathrm{R}}_{\mathrm{n}}$$

(14)

where ${\mathrm{R}}_{\mathrm{n}}$ is the net rate of production of species n by chemical reactions. An equation of this form will be solved for N − 1 species, where N is the total number of chemical species present in the system.

The EDC model is employed to represent the interaction between turbulence and the chemical reactions, which are assumed to occur inside the small turbulence structures [46,49,50]. The volume fraction of the fine scales and the reaction time scale are modeled, respectively, using the length scale (γ*) and the mean residence time (τ*).

$${\gamma }^{*}={\mathrm{C}}_{\gamma }{\left({\displaystyle \frac{\nu \epsilon }{{\mathrm{K}}^2}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$$

(15)

$${\tau }^{*}={\mathrm{C}}_{\tau }{\left({\displaystyle \frac{\nu }{\epsilon }}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$

(16)

In order to obtain a reliable representation of the combustion process, the DRM 19 chemical kinetic mechanism is adopted [51,52]. This reduced mechanism, derived from the detailed GRIMECH mechanism, consists of 19 species and 84 reversible reactions, along with the corresponding thermodynamic and transport properties. Additional details of the chemical kinetic mechanism are provided in a previous work [42]. The in situ adaptive tabulation (ISAT) method is utilized to optimize computational efficiency; the accuracy of this method was verified by comparing results obtained with a tolerance reduced by two orders of magnitude, confirming minimal deviation. The fluid mixture is treated as an incompressible ideal gas. The mixture-specific heat, molecular viscosity, and thermal conductivity are determined using the mixing law, while mass diffusivity is computed based on kinetic theory, ensuring consistency with fundamental transport principles.

This model was developed and validated through a simulation campaign on literature test cases in previous works [42,43].

3.1.5. Combustion Space Boundary Conditions

In the combustion chamber, a uniform airflow was introduced through the air vents at a fixed preheated temperature. At the burner inlets, the fuel mass flow, composed of natural gas, was supplied at ambient temperature. The exhaust gas port was assigned an outlet condition at atmospheric pressure. The furnace walls were modeled as no-slip boundaries, incorporating the available heat flow values (given from the glass producer) in various areas and an emissivity (ε_v) of 0.8. In

Table 1. Boundary conditions in the form of equations.

Equation	Inlet	Outlet	Wall
Momentum	${\rho \mathrm{v}}_{\mathrm{n}}={\displaystyle \frac{\dot{\mathrm{m}}}{\mathrm{S}}}$	${\mathrm{p}}_{\mathrm{atm}}$ ${\displaystyle \frac{{\partial \mathrm{u}}_{\mathrm{i}}}{\partial \mathrm{x}}=0}$	${\mathrm{u}}_{\mathrm{i}}=0$ Standard wall functions
Energy	${\mathrm{T}}_{\mathrm{in}}$	-	${\mathrm{T}}_{\mathrm{w}}={\displaystyle \frac{{\mathrm{q}}_{\mathrm{w}}-{\mathrm{q}}_{\mathrm{rad}}}{{\mathrm{h}}_{\mathrm{f}}}}+{\mathrm{T}}_{\mathrm{f}}$ Standard wall functions
Transport species	${\mathrm{Y}}_{\mathrm{n},\mathrm{in}}$	-	-
Turbulent kinetic energy	$\mathrm{K}=\frac{3}{2}{\left(\stackrel{\mathrm{-}}{\mathrm{u}} \mathrm{I}\right)}^2$ $\mathrm{I}={\displaystyle \frac{{\mathrm{u}}^{\prime }}{\stackrel{\mathrm{-}}{\mathrm{u}}}}$= 5%	-	${\displaystyle \frac{\partial \mathrm{K}}{\partial \mathrm{n}}}=0$
Turbulent dissipation rate	$\epsilon =0.09\rho {\displaystyle \frac{{\mathrm{K}}^2}{\mu }}{\left({\displaystyle \frac{{\mu }_{\mathrm{t}}}{\mu }}\right)}^{-1}$ ${\displaystyle \frac{{\mu }_t}{\mu }}=10$	-	$\mathrm{standard} \mathrm{wall} \mathrm{functions}$
Thermal radiation	${\epsilon }_{\mathrm{v}}$	${\epsilon }_{\mathrm{v}}$	${\epsilon }_{\mathrm{v}}$

, the boundary conditions, corresponding to each balance equation, are summarized.

3.2. Glass Tank

3.2.1. Glass Tank Model and Properties

The molten glass flow, dominated by free convection cells, was simulated as laminar flow, without radiation participation, by solving the mass, momentum, and energy equations. The glass properties were modeled to vary with the temperature according to step functions or polynomial relations in the appropriate temperature range.

Finally, the law of viscosity, with an expression linked to the chemical composition of the glass, had the following form:

$${\log }_{10}\mu =\mathrm{A}+{\displaystyle \frac{\mathrm{B}}{\mathrm{T}-{\mathrm{T}}_0}}$$

(17)

where the coefficients A, B, and T₀ are defined on the basis the data related to the specific production plant.

3.2.2. Glass Tank Boundary Conditions

At the doghouse inlet, the glass flow rate was set at ambient temperature, matching the daily production of the reference furnace. A suitable outlet condition was applied at the throat. The lateral walls and the furnace sole were treated as no-slip boundaries with an outgoing heat flow. To consider the thermal energy needed for the chemical reactions involved in glass melting and the precise proportion of the recycled material, a uniform negative heat source was incorporated into the domain.

4. Method of Solution

The computational domain includes both the combustion chamber and the molten glass bath, with each region discretized using an unstructured mesh composed of tetrahedral elements, created in Ansys ICEM CFD v. 17.1. In the combustion chamber, which encompasses the air and exhaust vents, as well as the fuel burner housings, a range of mesh densities was applied. Near the walls, the mesh was refined with prism layers to maintain a y+ value of around 30, ensuring the proper activation of the wall functions. For the glass bath, the mesh was created to align the elements on the glass surface with those in the domain above. Starting from this surface, prism layers were added to accurately capture the thermal gradient within the molten glass. The combustion chamber mesh consists of approximately 4 million elements, while the glass bath mesh contains 1.7 million cells. These grid configurations were selected based on a mesh sensitivity analysis carried out in previous studies [34,39,40]. A visual representation of the meshes in both domains is shown in Figure 2.

The simulations are carried out using the finite volume method code ANSYS Fluent 17.1 [53], employing a pressure-based solver with the coupled scheme for pressure–velocity coupling. The advection terms are discretized using a second-order upwind scheme.

The solution convergence is monitored through multiple criteria: residuals, monitor points, and global balances. Monitor points tracking key flow variables (such as temperature, velocity, and O₂ mole fraction) are placed along the fuel jet axis to verify the solution’s stability. A slight instability remains on the combustion side, as is typically encountered when steady RANS are performed on unstable flows [54,55]; this is further emphasized by combustion, as can be seen in real furnaces [31].

In conclusion, the observed instabilities suggest that the CFD model can be considered acceptable without the need to justify the use of additional stabilization or damping techniques [56].

Coupling of Combustion Space to Glass Tank Model

The methodology developed in this study aims to achieve a complete coupling between the combustion chamber and the glass bath, ensuring congruence on the free surface between heat flow and temperature. The procedure involves different steps.

Initially, the two domains are solved separately to provide a suitable initialization for the coupled simulation. For these initial simulations, the boundary condition at the free glass surface is set based on the solution of the adjacent domain. Then, the final innovation step involves the simultaneous simulation of both domains, initiated from the previous solutions, allowing for full thermal coupling. Between the combustion space and the glass bath, an interface is set with the thermally coupled wall option, imposing the appropriate emissivity on the combustion side. The two domains converge with their own timestep, with the glass timescale being much larger than that of the reactive flow. The effectiveness of the approach has been tested on simplified models including two domains with different fluid properties and velocities. The convergence of the two domains, starting from the preceding simulations, is reached within reasonable timeframes. This approach ensures that the overall thermal balance of the entire furnace is settled, providing a correct temperature and heat flow field on the glass that depends on both combustion and convective structures.

5. Results

This section presents the primary outcomes derived from the glass furnace model as applied to the industrial case described above. Comparisons with experimental measurements are shown, providing a reference for the procedure evaluation. In Figure 3, the volumetric rendering of OH mass fraction in the combustion chamber is depicted as the primary indicator of the structure of the flames.

This visualization reveals the development of two diffusive flames, generated by the intricate interaction between the preheated combustion air from the two towers and the fuel jets from the burners. The presence of flames is evident in the regions with the highest OH radical concentrations, particularly in the zone just downstream of the burners, extending to nearly half the furnace length. Figure 4 displays the streamlines within the combustion chamber, originating from the air vents and burners, with colors indicating temperature variations.

As mentioned above, the flow field exhibits a slightly unstable behavior, causing an alternate periodical retraction/elongation of the two flames, in response to the local flow variations; however, on average, the overall flow structure remains consistent with the description provided. These visualizations show the combustion-driven flow patterns, offering valuable insights into flame structures. It is evident that a significant interaction is present between the flames and the combustion products returning towards the exhaust gas tower, which is positioned higher than the two air inlets. This behavior, typical also of other glass furnace geometries, underlines the importance of using the adopted combustion model, whereas simplified models are intrinsically unsuitable.

For a detailed visualization, Figure 5 presents the temperature contours in the combustion chamber (with superimposed the velocity vectors) and glass bath, depicted in two vertical planes along the axes of the two flames. These distributions prominently highlight the effective coupling of the two domains, emphasizing the correspondence between the structure of the two flames and the glass temperature distribution. The proper thermal gradient within the molten glass, originating from the free glass surface, is accurately depicted. Specifically, the initial glass layer exhibits temperatures of approximately 1850 K, gradually decreasing by approximately 200 degrees towards the sole. At this particular instant, the slightly longer flame on the left side is influenced not only by the flow field fluctuations but also by the asymmetric position of the doghouse.

The primary achievement of the fully coupled simulation lies in its ability to accurately calculate the temperature field and heat flow at the interface. To illustrate this point, in Figure 6, contours are depicted on the glass surface, offering a comprehensive view of the thermal behavior in this critical region.

Both representations offer a clear depiction of the thermal impact of the flames on the glass. Additionally, upon comparing the previous contours, it becomes evident that the region exhibiting the highest heat flow corresponds to the hottest area. This high-temperature zone extends from the initial section of the furnace up to approximately half of the bath, gradually diminishing in the refinement area. A marked asymmetry is noticeable on the glass surface due to the inlet region of cold glass, which significantly absorbs a considerable amount of heat flow. Higher heat flows are visible in the areas near the salt line, where the corrosion of the higher walls is reflected in the higher measured power loss that is used as a boundary condition. Figure 7 presents the averaged radiation temperature distributions on the interface along the furnace axis. Values are obtained by a transverse average determined on the right/left of the axis, as well as on the entire width.

The temperature distribution analysis reveals a rapid increase in temperature within the initial region of the melting zone, with the maximum temperature observed at 43% of the furnace length. Beyond the halfway point, there is a consistent and nearly linear decrease in temperature. The observed temperature trend, initially rising and then decreasing, is due to the heat transfer dynamics and flame behavior. The temperature increases as the glass surface heats up from thermal radiation and direct interaction with the flame. However, after reaching a certain point, the temperature drops because the flame affects only the first half of the surface, reducing the heat flow to the rest of the glass as the flame moves away or weakens. Additionally, the asymmetry between the right and left regions is evident. Specifically, the slight differences in the peaks can be attributed to the fact that at this instant, the flame on the left side is more intense than the one on the right due to the inherently unstable nature of the flames. Furthermore, the presence of the doghouse on the right side lowers the temperature on that side as it absorbs a larger amount of heat power from the right flame.

Figure 8 illustrates the temperature contours on the walls of the furnace’s superstructure, including the crown and towers. In parallel, Figure 9 presents the temperature distribution on both the right and left superstructure compared with specific experimental measurement points for reference.

Furthermore, these depictions allow for the clear observation of the peak temperature region within the crown and superstructure, situated just before the midpoint of the furnace, corresponding to the previously identified flames, as previously explained. The numerical values derived from the simulation align well with the experimental measurements, accurately representing the overall temperature trend. A maximum error of about 1.5% has been detected. The combustion model has been validated in previous studies on CH₄ diffusive flames with a high-temperature oxidizer [42] and on an experimental IFRF furnace [43]. This validation steps are necessary due to the inherent difficulties of performing detailed measurements in industrial glass furnaces. So, typically, the complete furnace model validation relies on temperature measurements at specific points using thermocouples, as in this case. Building on the proven reliability of the combustion model, an innovative thermal coupling procedure is introduced, overcoming the simplifying assumptions and limitations of other approaches in the literature. This advancement enables a more accurate assessment of the CFD model’s reliability.

Figure 10 provides a visual representation of the temperature contours in the walls of the glass bath.

In this scenario, the temperature profile reveals a gradual decrease from the glass surface down to the sole. The coldest regions are situated in the vicinity before the weir wall and near the doghouse. Conversely, within the refining zone, elevated temperature values are evident, corresponding to the successive stages of glass melting and refining.

At the throat, the glass temperature obtained from the CFD simulation is 1726 K, while the experimental measurement is 1740 K. This comparative analysis reaffirms the precision of the developed numerical model, demonstrating its high accuracy when juxtaposed with experimental data. For a visual representation, Figure 11 illustrates the temperature distribution in the middle plane of the glass tank, with overlaid velocity vectors for added insight.

The convective motions of the melted glass reveal a distinct organizational pattern characterized by a macro-recirculation in the area preceding the weir wall, following an anticlockwise direction. Notably, on the free surface, the glass exhibits a back motion towards the flames. This recirculation is induced by the combined effects of the thermal gradient and the constriction caused by the wire wall positioned before the refining area. Within this refining region, the anticlockwise recirculation and a deeper underlying area are evident, facilitating the direct flow of molten glass into the throat.

Equations (18) and (19) outline the power balance in both the combustion chamber and the glass bath.

$${\mathrm{Q}}_{\mathrm{fuel}}+{\mathrm{Q}}_{\mathrm{air}}-{\mathrm{Q}}_{\mathrm{exhaust}}-{\mathrm{Q}}_{\mathrm{wall},\mathrm{cc}}={\mathrm{Q}}_{\mathrm{to}-\mathrm{glass}}$$

(18)

$${\mathrm{Q}}_{\mathrm{to}-\mathrm{glass}}={\mathrm{Q}}_{\mathrm{glass}, \mathrm{in}}-{\mathrm{Q}}_{\mathrm{glass},\mathrm{out}}-{\mathrm{Q}}_{\mathrm{wall},\mathrm{gt}}-{\mathrm{Q}}_{\mathrm{fusion}}$$

(19)

In the combustion chamber, Q_fuel is the product of the fuel mass flow rate and the corresponding low heating value; Q_air and Q_exhaust are computed by multiplying, respectively, the inlet air and the exhaust gas mass flow rate by the corresponding sensible enthalpy; and Q_wall,cc refers to the heat flow through the combustion chamber walls via thermal conduction.

In the glass bath, Q_to-glass is the heat flow from the combustion chamber, predominantly due to thermal radiation. At the glass bath exit, the power Q_glass,out exits through the throat, calculated as the product of the melted glass flow rate and its sensible heat, similarly to Q_glass,in, which is the power that enters from the doghouse. Within the glass bath, Q_wall,gt represents the power wasted through the walls for thermal conduction, while Q_fusion represents the power required for the transition of the solid glass into its liquid state.

In

Table 2. Average main contributions to the power balance.

Contribution	Value [%]
Qexhaust	−89.0
Qair	22.7
Qwall,cc	−7.5
Qto-glass	27.7
Qglass,out	−19.2
Qwall,gt	−5.3

, the different contributions, averaged on 1000 iterations, are reported as a percentage of Q_fuel.

The analysis of the power balance, considering both incoming and outgoing contributions, has confirmed the overall energy equilibrium of the system, thereby validating the stability and convergence of the simulation.

One of the key outcomes of the simulation is the quantification of heat flow directed towards the glass. Employing a fully coupled approach ensures a high degree of reliability in determining this value. This power, constituting just under a third of the power released from combustion, provides a substantial and favorable energy margin for the efficient melting of glass.

The simulation has allowed for an accurate evaluation of the chemical composition at the combustion chamber exit, facilitating the quantification of the main pollutants.

Table 3. Average main species of exhaust gases at the outlet.

Species	Value
O2 [%]	0.4
CO [ppm]	2457
H2O [%]	18.8
CO2 [%]	9.8

presents a summary of the main species of the exhaust gas at the outlet.

This composition reflects the typical emissions of combustion products for a glass furnace that operates with an air defect. Furthermore, the oxygen value found is perfectly in line with that which was measured experimentally, i.e., 0.5%.

6. Conclusions

In this study, a CFD model has been developed, with two innovative features, to accurately simulate a real industrial glass furnace. The combustion chamber is solved using the multispecies model with reduced kinetics based on the GriMech mechanism and the EDC turbulence–chemistry interaction model. The combustion and the glass domains, usually solved separately and coupled iteratively using temperature and heat flow as coupling variables, are solved simultaneously, allowing for a full thermal coupling. This approach resulted in a highly realistic flame structure, i.e., in accurate and consistent distributions of the heat flow and temperature on the glass surface, including their impact on the convective motions within the molten glass.

The model was used to simulate an existing glass furnace, providing detailed information on the reactive flow field, as well as on the glass domain. Experimental measurements, including the wall temperature on the crown, the glass temperature at the throat, and the oxygen concentration in the exhaust gases, have been compared with simulation results, confirming the validity of the model. Additionally, the power balance has been shown to be consistent.

This model has shown that the two diffusive flames extend over nearly half the furnace length, with the peak temperature on the free glass surface occurring at 43% of the total length. The temperature distribution on the superstructure exhibited a maximum deviation of just 1.5% from the experimental values. Moreover, a peak pattern in temperature was observed, driven by the flame flow structure. Regarding the bath walls, the results confirmed that in the refinement zone (after the split wall), higher temperatures are reached due to convective motions within the molten glass. The power balance analysis revealed that heat transfer from the combustion chamber, primarily through thermal radiation, accounts for approximately 28% of Q_fuel. Finally, the analysis of exhaust gas composition indicated a typical combustion emission for a glass furnace operating with an air defect, characterized by a very low oxygen concentration (0.4%).

The primary goal of this model is to serve as a robust tool for designing glass furnaces with an emphasis on energy optimization. It enables subsequent parametric analyses based on different fuels and geometric parameters, fostering a comprehensive understanding of the system for enhanced energy efficiency.